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Showing papers on "Continuous automaton published in 2017"


Proceedings ArticleDOI
18 Apr 2017
TL;DR: A new sampling-based algorithm that builds incrementally a directed tree that approximates the state-space and transitions of the synchronous product automaton is proposed that is probabilistically complete and asymptotically optimal and can be used to model-check product automata with billions of states.
Abstract: This paper proposes a sampling-based algorithm for multi-robot control synthesis under global Linear Temporal Logic (LTL) formulas. Robot mobility is captured by transition systems whose states represent regions in the environment that satisfy atomic propositions. Existing planning approaches under global temporal goals rely on graph search techniques applied to a synchronous product automaton constructed among the robots. As the number of robots increases, the state-space of the product automaton grows exponentially and, as a result, graph search techniques become intractable. In this paper, we propose a new sampling-based algorithm that builds incrementally a directed tree that approximates the state-space and transitions of the synchronous product automaton. By approximating the product automaton by a tree rather than representing it explicitly, we require much fewer resources to store it and motion plans can be found by tracing the sequence of parent nodes from the leaves back to the root without the need for sophisticated graph search techniques. This significantly increases scalability of our algorithm compared to existing model-checking methods. We also show that our algorithm is probabilistically complete and asymptotically optimal and present numerical experiments that show that it can be used to model-check product automata with billions of states, which was not possible using an off-the-shelf model checker.

21 citations


Journal ArticleDOI
TL;DR: The partial derivative automaton and the position automaton are generalized to regular expressions with shuffle, and their state complexity in the worst, as well as in the average case is studied.
Abstract: We generalize the partial derivative automaton and the position automaton to regular expressions with shuffle, and study their state complexity in the worst, as well as in the average case. The number of states of the partial derivative automaton ( A p d ) is, in the worst case, at most 2 m , where m is the number of letters in the expression. The asymptotic average is bounded by ( 4 3 ) m . We define a position automaton ( A p o s ) that is homogeneous, but in which several states can correspond to a same position, and we show that A p d is a quotient of A p o s . The number of states of the position automaton is at most 1 + m ( 2 m − 1 ) , while the asymptotic average is no more than m ( 4 3 ) m .

17 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the word problem for automaton semigroups and automaton groups from a complexity point of view, and showed that the uniform word problem is NP-hard.

14 citations


Book ChapterDOI
TL;DR: This work shows how a computation can be performed on the collider by exploiting interactions between gliders (particles, localisations) and proposes constructions based on universality of elementary cellular automaton rule 110, cyclic tag systems, supercolliders, and computing on rings.
Abstract: A cellular automaton collider is a finite state machine build of rings of one-dimensional cellular automata. We show how a computation can be performed on the collider by exploiting interactions between gliders (particles, localisations). The constructions proposed are based on universality of elementary cellular automaton rule 110, cyclic tag systems, supercolliders, and computing on rings.

13 citations


Journal ArticleDOI
TL;DR: The collective interaction of agents for jointly overcoming (negotiating) obstacles is simulated using a cellular automaton, which is based on the probabilities of going from one cell to another.
Abstract: The collective interaction of agents for jointly overcoming (negotiating) obstacles is simulated. The simulation uses a cellular automaton. The automaton’s cells are filled with agents and obstacles of various complexity. The agents' task is to negotiate the obstacles while moving to a prescribed target point. Each agent is assigned to one of three levels, which specifies a hierarchy of subordination between the agents. The complexity of an obstacle is determined by the amount of time needed to overcome it. The proposed model is based on the probabilities of going from one cell to another.

12 citations


Journal ArticleDOI
26 Oct 2017-Chaos
TL;DR: Two statistically stationary states with power-law scaling of avalanches are found in a simple 1 D cellular automaton and the migration of states of the automaton between these two self-organized criticality states is demonstrated during evolution of the system in computer simulations.
Abstract: Two statistically stationary states with power-law scaling of avalanches are found in a simple 1 D cellular automaton. Features of the fixed points, the spiral saddle and the saddle with index 1, are investigated. The migration of states of the automaton between these two self-organized criticality states is demonstrated during evolution of the system in computer simulations. The automaton, being a slowly driven system, can be applied as a toy model of earthquake supercycles.

5 citations


Journal ArticleDOI
TL;DR: It is proved that it remains NP-complete even if restricted to Eulerian automata with binary alphabets as it has been conjectured by Martyugin (2011).
Abstract: A word is called a reset word for a deterministic finite automaton if it maps all the states of the automaton to a unique state. Deciding about the existence of a reset word of a given length for a given automaton is known to be an NP-complete problem. We prove that it remains NP-complete even if restricted to Eulerian automata with binary alphabets as it has been conjectured by Martyugin (2011).

4 citations


Journal ArticleDOI
TL;DR: It is proved that a non-abelian free group can be generated by a two-state bi-reversible automaton over a changing alphabet X=(Xi)i1 if and only if X is unbounded.

3 citations


01 Jan 2017
TL;DR: It is demonstrated that an explicit expression "solution of the CA" can be used to obtain an expected value of a given cell after $n$ iterations, provided that the initial condition is drawn from a Bernoulli distribution.
Abstract: Cellular automata (CA) are fully discrete alternatives to partial differential equations (PDE). For PDEs, one often considers the Cauchy problem, or initial value problem: find the solution of the PDE satisfying a given initial condition. For many PDEs of the first order in time, it is possible to find explicit formulae for the solution at the time $t>0$ if the solution is known at $t=0$. Can something similar be achieved for CA? We demonstrate that this is indeed possible in some cases, using elementary CA rule 172 as an example. We derive an explicit expression for the state of a given cell after $n$ iteration of the rule 172, assuming that states of all cells are known at $n=0$. We then show that this expression ("solution of the CA") can be used to obtain an expected value of a given cell after $n$ iterations, provided that the initial condition is drawn from a Bernoulli distribution. This can be done for both finite and infinite lattices, thus providing an interesting test case for investigating finite size effects in CA.

2 citations


04 Sep 2017
TL;DR: The article presents the results of the simulation of a cellular automaton model whose cells assume continuous values, aimed at finding an answer to the question whether there are any general regularities or patterns describing the dynamics of the assumed model.
Abstract: The article presents the results of the simulation of a cellular automaton model whose cells assume continuous values. The described experiment examined the behaviour of local entropy and showed the occurrence of its three qualitatively different images with their character resembling Turing patterns. The hypothesis, according to which the behaviour of local entropy might be identified with an image created in a reaction-diffusion process, was presented. Problem The research study concerning the mathematical foundations of creating cultures (Zgrzebnicki, in press-a) considers a continuous cellular automaton model as a hypothetical framework. Although there have been many studies on the properties of discrete cellular automata published so far, the models whose cells assume any values limited only by floating point precision have been much less discussed in scientific works. The research presented herein assumed that the discussed automaton may change values of its cells according to any function transforming the values of neighbours in the neighbourhood of a certain radius, and that this radius and weight of the neighbours' contribution to a cell's change may also be arbitrarily defined for the purposes of a certain experiment. In comparison with discrete automata (Wolfram, 1984), such assumptions imply a much broader class of solutions. The conducted research is aimed at finding an answer to the question whether there are any general regularities or patterns describing the dynamics of the assumed model. Method Let there exist a matrix !, whose components are called cells. Let there exist a function ", which transforms matrix ! into matrix !’ in a way that: $% ′ = "()% * $*, ,) , where $ stands for the cell's value, while % and * mean coordinates, and r means a radius within which the neighbourhood of a certain cell has an influence on its value's transformation. Furthermore, to eliminate the problem of boundary values, let us assume that the boundaries of the matrix are glued together so that topology of the described space is identical with a torus. Figure 1: The value of cells established on the basis of the values of their neighbours located within the radius , = 3. Topology of glued boundaries guarantees continuity of solutions on the edges of the matrix. Finally, let us consider process 0 in which values of all the cells are changed so that a matrix obtained after the change becomes a source matrix for the next step: 0:! → !3 → !33 → !333 → ⋯

2 citations


Posted Content
TL;DR: The Sayab-rule as mentioned in this paper is a binary 2D cellular automaton with a Moore neighborhood and isotropic dynamics, consisting of just four live cells at its minimal phases.
Abstract: To understand the underlying principles of self-organisation and computation in cellular automata, it would be helpful to find the simplest form of the essential ingredients, glider-guns and eaters, because then the dynamics would be easier to interpret. Such minimal components emerge spontaneously in the newly discovered Sayab-rule, a binary 2D cellular automaton with a Moore neighborhood and isotropic dynamics. The Sayab-rule has the smallest glider-gun reported to date, consisting of just four live cells at its minimal phases. We show that the Sayab-rule can implement complex dynamical interactions and the gates required for logical universality.

Journal Article
TL;DR: In this paper, a homogeneous systolic pyramid automata with four-dimensional layers (4-HSPA) is proposed, which is a parallel model of various cellular automata.
Abstract: Cellular automaton is famous as a kind of the parallel automaton. Cellular automata were investigated not only in the viewpoint of formal language theory, but also in the viewpoint of pattern recognition. Cellular automata can be classified into some types. A systolic pyramid automata is also one parallel model of various cellular automata. A homogeneous systolic pyramid automaton with four-dimensional layers (4-HSPA) is a pyramid stack of four-dimensional arrays of cells in which the bottom four-dimensional layer (level 0) has size an (a≥1), the next lowest 4(a-1), and so forth, the (a-1)st fourdimensional layer (level (a-1)) consisting of a single cell, called the root. Each cell means an identical finite-state machine. The input is accepted if and only if the root cell ever enters an accepting state. A 4-HSPA is said to be a real-time 4-HSPA if for every four-dimensional tape of size 4a (a≥1), it accepts the fourdimensional tape in time a-1. Moreover, a 1- way fourdimensional cellular automaton (1-4CA) can be considered as a natural extension of the 1-way two-dimensional cellular automaton to four-dimension. The initial configuration is accepted if the last special cell reaches a final state. A 1-4CA is said to be a real- time 1-4CA if when started with fourdimensional array of cells in nonquiescent state, the special cell reaches a final state. In this paper, we proposed a homogeneous systolic automaton with four-dimensional layers (4-HSPA), and investigated some properties of real-time 4-HSPA. Specifically, we first investigated the relationship between the accepting powers of real-time 4-HSPA’s and real-time 1-4CA’s. We next showed the recognizability of four-dimensional connected tapes by real-time 4-HSPA’s.

Posted Content
01 Jul 2017-viXra
TL;DR: In this paper, a cellular automaton molecular model based on the classical wave equation was proposed as an alternative to cellular automata-based QM. This is a fresh approach started by some authors including Prof. Gerard ‘t Hooft.
Abstract: In a recent paper, it has been argued that QM can arise from classical cellular automata. This is a fresh approach started by some authors including Prof. Gerard ‘t Hooft. Nonetheless, in a previous paper, we have reviewed some inadequacies of Schrodinger equation, hence the entire wave mechanics. According to Shpenkov, the classical wave equation is able to derive a periodic table of elements -which is close to Mendeleyev’s periodic table-, and also other phenomena related to the structure of molecules. It is suggested that Shpenkov’s interpretation of classical wave equation can complement Schrodinger equation. Therefore in this paper we will discuss how we can arrive to a cellular automaton molecular model starting from classical wave equation, as an alternative to cellular automata based QM.

Journal ArticleDOI
TL;DR: In this article, the random walk method is employed to deal with this problem in two steps: first, the velocity potential is determined as a steady-state solution, and the modified concentration, i.e., the product of the velocity and the concentrations of the oxidized and reduced chemical species, is calculated with the simultaneous application of the electrostatic potential.
Abstract: One-dimensional electrochemical cellular automata have been extended to incorporate a system of 300 × 300 cells for considering the flow of an electrolytic solution. The random-walk method is employed to deal with this problem in two steps: First, the velocity potential is determined as a steady-state solution. Second, the modified concentration, i.e., the product of the velocity potential and the concentrations of the oxidized and reduced chemical species, is calculated with the simultaneous application of the electrostatic potential. The model is applied to the calculation of the Cottrell current, or the transient response to the step voltage, and the results indicate that the limiting currents linearly increase with increasing flow velocity. © The Electrochemical Society of Japan, All rights reserved.

Journal ArticleDOI
TL;DR: A finite automaton based algorithm for identification of infinite clusters in a 2D rectangular lattice with cells that has a computational complexity of and could be appropriate for efficient data flow implementation.
Abstract: We propose a finite automaton based algorithm for identification of infinite clusters in a 2D rectangular lattice with cells. The algorithm counts infinite clusters and finds one path per infinite cluster in a single pass of the finite automaton. The finite automaton is minimal according to the number of states among all the automata that perform such task. The correctness and efficiency of the algorithm are demonstrated on a planar percolation problem. The algorithm has a computational complexity of and could be appropriate for efficient data flow implementation.

Journal ArticleDOI
TL;DR: In this paper, it was shown that there is a weakly universal cellular automaton on the pentagrid with two states and that the set of non quiescent states has always infinitely many cycles.
Abstract: In this paper, we prove that there is a weakly universal cellular automaton on the pentagrid with two states. This paper improves in some sense a previous result with three states. Both results make use of a la Moore neighbourhood. However, the result with three states is rotation invariant while the result of the present paper is not. In both cases, at each step of the computation, the set of non quiescent states has always infinitely many cycles.

Journal ArticleDOI
TL;DR: In this paper, the integrability properties of a reversible deterministic cellular automaton were studied and the decay modes in the first orbital (containing the leading decay mode) were constructed in terms of an exact inhomogeneous matrix product ansatz.
Abstract: We study integrability properties of a reversible deterministic cellular automaton (the rule 54 of [Bobenko et al, Commun Math Phys 158, 127 (1993)]) and present a bulk algebraic relation and its inhomogeneous extension which allow for an explicit construction of Liouvillian decay modes for two distinct families of stochastic boundary driving The spectrum of the many-body stochastic matrix defining the time propagation is found to separate into sets, which we call orbitals, and the eigenvalues in each orbital are found to obey a distinct set of Bethe-like equations We construct the decay modes in the first orbital (containing the leading decay mode) in terms of an exact inhomogeneous matrix product ansatz, study the thermodynamic properties of the spectrum and the scaling of its gap, and provide a conjecture for the Bethe-like equations for all the orbitals and their degeneracy

Posted Content
TL;DR: A Novel Automaton Model which uses Arithmetic Operations as the Evolving Rules, each cell has the states of the Natural Numbers k = (N), a radius of r = 1/2 and operates on an arbitrary input size is proposed.
Abstract: We Propose A Novel Automaton Model which uses Arithmetic Operations as the Evolving Rules, each cell has the states of the Natural Numbers k = (N), a radius of r = 1/2 and operates on an arbitrary input size. The Automaton reads an Arithmetic Expression as an input and outputs another Arithmetic Expression. In Addition, we simulate a variety of One Dimensional Cellular Automata Structures with different Dynamics including Elementary Cellular Automata.

Posted Content
TL;DR: In this paper, a 3-state cellular automaton with degenerate hyperbolicity was presented, and the densities of 0, 1 and 2 after n iterations of this rule were calculated using finite state machines.
Abstract: In a recent paper [arXiv:1506.06649 [nlin.CG]], we presented an example of a 3-state cellular automaton which exhibits behaviour analogous to degenerate hyperbolicity often observed in finite-dimensional dynamical systems. We also calculated densities of 0, 1 and 2 after n iterations of this rule, using finite state machines to conjecture patterns present in preimage sets. Here, we re-derive the same formulae in a rigorous way, without resorting to any semi-empirical methods. This is done by analysing the behaviour of continuous clusters of symbols and by considering their interactions.